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Maximal estimates for the $(C,\alpha )$ means
of $d$-dimensional Walsh-Fourier series

Author: Ferenc Weisz
Journal: Proc. Amer. Math. Soc. 128 (2000), 2337-2345
MSC (1991): Primary 42C10, 43A75; Secondary 60G42, 42B30
Published electronically: November 29, 1999
MathSciNet review: 1664379
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Abstract: The $d$-dimensional dyadic martingale Hardy spaces $H_{p}$ are introduced and it is proved that the maximal operator of the $(C,\alpha )$ $(\alpha =(\alpha _{1},\ldots ,\alpha _{d}))$ means of a Walsh-Fourier series is bounded from $H_{p}$ to $L_{p}$ $(1/(\alpha _{k}+1)<p<\infty )$ and is of weak type $(L_{1},L_{1})$, provided that the supremum in the maximal operator is taken over a positive cone. As a consequence we obtain that the $(C,\alpha )$ means of a function $f \in L_{1}$ converge a.e. to the function in question. Moreover, we prove that the $(C,\alpha )$ means are uniformly bounded on $H_{p}$ whenever $1/(\alpha _{k}+ 1)<p < \infty $. Thus, in case $f \in H_{p}$, the $(C,\alpha )$ means converge to $f$ in $H_{p}$ norm. The same results are proved for the conjugate $(C,\alpha )$ means, too.

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Additional Information

Ferenc Weisz
Affiliation: Department of Numerical Analysis, Eötvös L. University, H-1117 Budapest, Pázmány P. sétány 1/D, Hungary
Address at time of publication: Department of Mathematics, Humboldt University, D-10099 Berlin, Unter den Linden 6, Germany

Keywords: Martingale Hardy spaces, $p$-atom, atomic decomposition, $p$-quasi-local operator, interpolation, Walsh functions, $(C, \alpha )$ summability
Received by editor(s): September 16, 1998
Published electronically: November 29, 1999
Additional Notes: This research was done while the author was visiting the Humboldt University in Berlin and was supported by the Alexander von Humboldt Foundation.
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2000 American Mathematical Society

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